Data AR(1)

St. John's

for

Longitudinal Data

by

@Manish Madan

A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirement for the Degree of

Master of Science in Statistics

Department of Mathematics and Statistics Memorial University of Newfoundland

August, 2006

Newfoundland Canada

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The statistical analysis of gamma data (exponential being a special case) is quite common in many biomedical or engineering research. The existing studies deal with this type of data either in the independence or time series set up. Independence, as the name suggests implies that the data at time point 't + 1' is independent of the data at time point 't', whereas the time series set up suggests dependence in the data collected at subsequent time points.

It may however happen in practice that one collects the gamma responses re- peatedly along with a set of multi-dimensional covariates, from a large number of independent individuals over a small period of time. In this set up, it is natural that the repeated gamma responses of an individual will be correlated. It is of interest to obtain consistent and efficient estimates for the effects of the covariates on the responses after taking the longitudinal correlation into account.

In this thesis, we study an autoregressive order 1 (AR(1)) type longitudinal gamma model consisting of a regression vector, a scale, and a longitudinal correlation parame- ter. The likelihood and a generalized quasilikelihood (GQL) inferences are considered for the estimation of these parameters. It is argued that the likelihood approach is extremely complicated whereas the GQL approach appears to be much simpler which also provides consistent and highly efficient estimates. This is verified through a simulation study.

ii

To give just an acknowledgment to Professor B.C. Sutradhar would be an understate- ment, for his role and plethora of guidance given to me during my M.Sc. program and towards the completion of my thesis has been immense. His guidance has been as vital and as important, as one may imagine. With this, I would like to express my deep heart-felt gratitude to my supervisor, Prof. B.C. Sutradhar for his supervi- sion, advice, encouragement, enduring patience and constant support. He has never been ceasing in his support in every possible way towards making me able, to ac- complish my first ever thesis. He has always provided clear explanations when I was

(hopelessly) stuck, constantly driving the research with new and innovative research methodologies (I wonder where does he get all this from?!). To begin my future pur- suits, I had to leave the University sooner than expected and he was very kind and gracious to me by increasing my research appointments with him and also, giving most of his own research time as well, thus making me able to finish the thesis in a very short and record span of time. I'd always feel immense gratitude for this favor in particular apart from many others (the list may indeed be long!). It was undoubtedly a great privilege for me to work under his supervision for which I shall forever be grateful. I thank him whole heartily.

I want to sincerely acknowledge the financial support provided by the School of Graduate Studies, Department of Mathematics and Statistics, Prof. B.C. Sutradhar in the form of Graduate Assistantship & Teaching Assistantships. Further, I wish to thank the Department, the lovely and helpful staff at our General Office for providing

iii

a very friendly ambience, nice conversations and the necessary facilities, especially the stationary material for which they have been very generous to me.

I also thank Dr. JC Loredo-Osti and Dr. Zhaozhi Fan of Department of Mathematics & Statistics, Memorial University of Newfoundland for taking the time to serve as examiners of this thesis. Their constructive comments and suggestions served to improve the quality of the thesis.

I personally thank Mr. Afework Solomon, for the eternal and inspiring wisdom that he has shared with me during his friendship with me. Words may not be just enough to thank him for what he has given me but let me assure, it will continue to hold a special place in my heart forever.

It is my great pleasure to thank all my friends and well-wishers who directly or indirectly encouraged and helped me in the M.Sc. program and contributed to this dissertation.

Last and most importantly, I am very much grateful to my parents, my elder brother, my sister-in-law and my younger brother, for their strong belief in my ca- pabilities and for all of their eternal love, support and encouragement during my stay far away from them.

Abstract

Acknowledgements

List of Tables

1 Introduction

1.1 Background of the Problem 1.2 Objective of The Thesis ..

2 Stationary AR( 1) Gamma Model

2.1 Time Series Models . . . .

2.1.1 The Gamma Autoregressive Process, GAR(1)

2.1.2 The Gamma Beta Autoregressive Process, GBAR(1) 2.2 Regression Models in Longitudinal Set up

2.2.1 Basie Properties of the Model: Mean, Variance, and Auto- Covariance Structure . . . .

vi

ii

iii

viii

1

1 2

3

3 4 5 6

8

3.1.1 Construction of the Likelihood Function 3.1.2 Estimation . . . .

3.2 Quasi- Likelihood Estimation 3.3 A Simulation Study . . . .

4 Non-stationary AR( 1) Type Longitudinal Gamma Models

5 Conclusion

A

Bibliography

13 15

22 24

30

33

35

53

3.1 Simulated means (SM) and Simulated standaTd eTTOTS (SSE) of the GQL estimates for regression parameters (3_{1 ,} /32 ; and the moment es- timates of the nuisance pammeters p and ~' for ~ = 0.5 and p =

0.2, 0.4, 0.5, 0.6, 0.8, 0.9 joT the longitudinal gamma AR(i) set up with K = ^{100, T} = 4, (31 = /32 = ^1.0, based on 5000 simulations . . . . 27 3.2 Simulated means (SM} and Simulated standard eTTOTs (SSE) of the

G Q L estimates JoT TegTession pammeteTs (3_{1 ,} {'3_{2 ;} and the moment es- timates of the nuisance pammeteTs p and ~' faT ~ = 1.0 and p =

0.2, 0.4, 0.5, 0.6, 0.8, 0.9 for the longitudinal gamma AR(i) set up with K = ^{100, T} = 4, /31 = /32 = 1.0, based on 5000 simulations . . . . 27 3.3 Simulated means (SM) and Simulated standard err-ors (SSE) of the

GQL estimates for Tegression parameters {3_{1 ,} f32;and the moment es- timates of the nuisance parameters p and !;,, for !;, = ^{1.5 and p} =

0.2, 0.4, 0.5, 0.6, 0.8, 0.9 for the longitudinal gamma AR(i) set up with K = 100, T = 4, (3₁ = /32 = 1.0, ba8ed on 5000 simulations . . . . 28

viii

Introduction

1.1 Background of the Problem

In traditional longitudinal studies it is common to collect binary or count responses repeatedly along with a set of multi-dimensional covariates over a small period of time from a large number of independent individuals. We, for example, refer to Liang and Zeger (1986), Crowder (1995), Sutradhar and Das (1999), and Sutradhar (2003) for such longitudinal studies. There has also been some discussion on the longitudinal studies for exponential failure time data. For this type of studies we refer to Cai and Prentice (1995), Guo and Lin (1994), Lin (1994), Hsu and Prentice (1996), Prentice and Hsu (1997), and Hasan (2004). Note that as opposed to the longitudinal set up, that is, in the independence set up, there exists many studies [see Kalbfleisch and Prentice (2002), for example] involving exponential, gamma and Weibull data, for example. But there is no adequate discussion in the literature on the use of gamma or Weibull data in the longitudinal set up. This motivated us to explore a longitudinal model for the gamma data.

As far as the inference in the longitudinal set up is concerned, we refer to the generalized quasilikelihood (GQL) approach suggested recently by Sutradhar (2003).

1

This GQL approach is treated to be an alternative simpler approach as compared to the maximum likelihood (ML) approach. In fact, in practice, it may not be possible to write the likelihood function for certain longitudinal models which makes the like- lihood approach useless. In contrast, recent studies in the longitudinal set up indicate that the GQL approach is quite simpler and it provides consistent and highly efficient estimates for the parameters of the longitudinal model. This motivated us to adapt the GQL estimation approach in our set up to make inferences about the parameters involved in our longitudinal gamma model.

1.2 Objective of The Thesis

One of the main objectives of the thesis is to develop an autoregressive order 1 (AR(l)) type longitudinal model for gamma data which we do in chapter 2 for the stationary gamma data. Here, stationarity means that the multi-dimensional covariates asso- ciated with the repeated gamma responses are time independent. This also leads to a longitudinal correlation structure which is independent of time. The second main objective of the thesis is to make inferences about the parameters of the longitudinal gamma model. In our set up, there will be 3 types of parameters: (i) the regression effects, (ii) a scale parameter and (iii) a longitudinal correlation parameter. The esti- mation of these parameters are discussed in chapter 3 for the longitudinal stationary model. As far as the estimation techniques are concerned, we consider the well known likelihood approach and a relatively less known GQL approach. We conduct a simu- lation study in the same chapter to examine the performance of the GQL approach, likelihood approach being extremely complicated. We also study the basic properties of a longitudinal non-stationary AR(l) type model in chapter 4. The thesis concludes in chapter 5.

Stationary AR(l) Gamma Model

In the traditional longitudinal set up [Sutradhar, 2003], one collects discrete such as Poisson or Binary data along with multi-dimensional covariates over a short period of time, say T, from a large number of independent individuals, say K. But, there are situations in practice where the response may follow continuous such as exponential [Hassan, 2004] or the gamma distribution. For the purpose, in the thesis, we will consider a longitudinal model for the responses following gamma distribution. Note that there does not exist any discussions in the literature on this type of longitudinal model (where T is small and K ---+ oo), there exists however time series model, for gamma data, that is, for the case when T ---+ oo and K=l.

In the following section, we present some of these time series models.

2.1 Time Series Models

The gamma distribution has found extensive application in reliability and life test- ing [see Engelhardt and Bain (1977), Glaser (1976), and Gross and Clark (1975), for example] and in insurance [see Ammeter (1970) and Seal (1969), for example].

3

The statistical inferences using such gamma distributions for example, have been dis- cussed by DiCiccio (1987), Lawless (1980), and Miller (1980). These inferences were, however, confined to the independence set up only. But, in practice, it may happen that the gamma responses are collected from an individual system over a long period of time. The responses in this case will be naturally correlated. Some authors such as Lewis (1982), and Gaver and Lewis (1980) have modeled this type of correlated gamma data. To be specific, these authors have studied the distribution properties of the the correlated gamma data in the time series set up. To have a feel for such models, we review in brief some of their models as in the following.

2.1.1 The Gamma Autoregressive Process, GAR(l)

Let {yt}, t = 1, ... , T be a sequence of responses collected over T time points. Also, let { dt} be a sequence of independent and identically distributed random variables.

For the cases when Yt follows a gamma distribution marginally, say Yt"' Ga(1, >.), that is

(2.1)

Gaver and Lewis (1980) [see also Lawrence (1982)] introduced an AR(1) model given by

Yt = a Yt-l + dt (2.2)

where a is a correlation parameter ranging from 0 to 1 whereas in the classical Gaus- sian model this type of parameter satisfies a wider range from -1 to + 1. Note that, it was shown by these authors that to maintain the same marginal gamma distribution for Yt for all t = 1, ... , T, it is essential that dt in (2.2) follows the distribution of a mixture given by

dt = _{ ⁰ with probability a

Gamma(!, .X) with probability 1-a(= a) (2.3)

For details on other distributional properties of this model (2.2), one may refer to the above mentioned studies.

2.1.2 The Gamma Beta Autoregressive Process, GBAR(l)

The gamma process mentioned in last subsection (2.1.1) was developed by Gaver and Lewis (1980) for the one parameter gamma family. An extension of this type of process to the two parameters gamma family is generally complicated because of the complexity of its innovation process. Lewis (1982) has provided a more flexible and simpler approach for gamma processes in general. To be specific, Lewis (1982) presented a linear, random coefficient auto-regression model

Yt = at Yt-t + dt (2.4)

where Yt-t ^rv Ga(.X, ~) and at has the beta distribution, namely, at "' Be(.Xt, A- At), that is,

_ l >.-t -/;Yt-1

f(Yt-t I .X,~) - r(.X) ~->. Yt-t e (2.5)

(0 <at < 1) (2.6)

. r(.Xt) r(.x- .At)

With Beta(At,A- At) = r(.X) .

Here, to maintain the same distribution for Yt as that of Yt-l, it is essential that dt in (2.4) follows the gamma distribution, namely, dt "' Ga(.X -At,~). It may be further noted that dt, Yt-t, and at are independent for all t. Furthermore, it was

shown by Lewis (1982) that the autocorrelation function of the process has the form given by Pv(k) = [E(at)]k = ()q ~ ^{.xJk, where ..\2 = ..\-..\1.}

2.2 Regression Models in Longitudinal Set up

In this section, we exploit the time series model considered by Lewis (1982) for gamma data in the longitudinal set up. To be specific, we write the model (2.4) for the longitudinal case as

Yit = ait Yi,t-1 + ^{dit (2.7)}

where, Yit denotes the response collected at time t (t = ^{1, ... , T) from the ith(i} =

1, ... , K) individual, ait and dit are similar variables as in (2.4) corresponding to time point t for a given i. Note that in (2.7), T is considered to be small and K --+ oo; whereas in (2.4), K = ^{1 and} T--+ oo in terms of the notations in (2.7).

In the time series set up such as (2.4), many authors confined their studies to the non-regression models. Since the responses recorded in a longitudinal set up are often affected by certain covariates, in this thesis, we are mainly interested to the inferences about the effects of such covariates after taking the correlations of the repeated gamma data into account.

Let xit = (xit1 , . . . , Xitu, ... , Xitp)' be the p-dimensional covariate vector corre- sponding to Yit and (3 = ((31, ... , (3p )' is the effect of X it on Yit for all t = ^{1, ... ,} T and all i = 1, ... , K. Thus, in notation, it is of main interest to estimate the regression effect (3 after taking the correlations of Yi1 , ... , Yit, ... , YiT into account.

Note that in general it is likely that Xit's are time dependent covariates. This time dependent case will be referred to as the non-stationary case which we will deal with, in chapter 4. However, when Xit remains fixed for all t = 1, ... , T, the non- stationary case reduces to the stationary case. In this section, we deal with this type

of stationary models.

For the stationary case, let Xi. = Xit for all t = ^{1, ... ,} T. FUrther, let

). . = z.1 e-x:./3 = e-(xi.1fJ1 + ... +xi.p/3p) and ). . z.2 = g(>.. ) z.1 ' ^(2.8)

where g(·) is a suitable known function. Now by following Lewis (1982), we provide the distributional result for the model (2.7) as in the following lemma.

Lemma 2.1. Suppose that~ > 0 is a scale parameter. If Yi,t-1 "' Ga(>.i.l + Ai.2, ~),

ait "' Be(>.i.l, >.i.2) and dit "' Ga(>.i.2, ~), and Yi,t- 1, ait and dit are assumed to be independent to each other, then Yit"' Gamma(>.i.1 + ^{>.i.2, ~).}

Proof. In (2.7), let Zit= ait Yi,t-1· Now, as ait"' Be(>.i.1,>.i.2), it follows from (2.6) that the probability density function (pdf) of ait is given by

!( I , , ) _ r(>.i.l + >.i.2) .xi.l-1 (₁ ).xi 2-1 CXit Ai.1, Ai.2 - r(.\. ) r(.\. ) (Xit - CXit . ·

z.1 z.2 ^(2.9)

Similarly, as Yi,t-1 "'Ga(>.i.1 + Ai.2, ~), the pdf of Yi,t-1 by (2.5) may be written as (2.10) Next, as Yi,t- 1 and ait are assumed to be independent, by using the transformation wi,t-1 = Yi,t-1 - Zit, where Zit = ait Yi,t-1, it then follows that the joint density of Zit and wi,t- 1 is given by

(2.11)

We now obtain the marginal distribution of Zit by integrating (2.11) over wi,t-1 as follows:

(2.12)

which is the pdf of gamma Zit· That is, Zit ^rv Gamma(>.i.1 , ~). Therefore, by using the additive property of the gamma distribution [Johnson and Kotz (1979)] for two independent gamma variable, we obtain the distribution of lit as Yit ^rv Ga(>.i.l +

>.i.2, ~).

In the following subsection, we provide the auto-covariance structure for the gamma responses under the model (2.7).

2.2.1 Basic Properties of the Model: Mean, Variance, and Auto-Covariance Structure

Let f.-lit and O'itt denote the mean and variance of Yit for all t = 1, ... , T, respectively.

By Lemma (2.1), these mean and variance are given by

(2.13) (2.14)

Following the time series model considered by Lewis (1982), we consider Pi = ).. Ai.\

i.l + i.2 which is the mean of ait in the stationary case. This formula helps one to identify the relationship between >.i.l and >.i.2 under this type of stationary model. That is, for given Pi, Ai.2 =

C

Using the above notations we now state the auto-correlations structure for {yit} as in the following lemma.

Lemma 2.2. For u < t, the (t-u)-th lag correlation between Yiu and Yit is given by corr(liu, lit) = p~-u

Proof. We prove this lemma by induction. For the purpose, we first find the lag 1 covariance, namely cov(lit, li,t-1),

Lag 1 auto-covariance:

cov(lit, li,t-1) =E(lit li,t-1) - E(lit) E(li,t-1)

=E[(ait li,t-1 + dit) li,t-1]- E(lit) E(li,t-1)

=Eait (ait Yi~t-1) + Ed;1 (dit li,t-1) - E(lit) E(li,t-1)

As ait has the beta distribution with parameters >.i.1 ^and >.i.2, it follows that E(ait) = ).. ).~\ . . Similarly, as Yi,t-1 ^{f ' . j} Ga(>.i.l +.Xi.2,e) and dit ^{f ' . j} Ga(>.i.2,e), it follows from

z.1 z.2

(2.13) and (2.14) that E(Yi~t-1⁾ = (>.i.1 + >.i.2) (;;·1 + '\.2 + ¹) and E(dit) = >.~.²

Furthermore, as ait, Yit and dit are independent, after some algebra we obtain

Next for convenience, we re-express the covariance in terms of Pi as

cov(lit, li,t-d ^>.i.1 >.i.1 + >.i.2 .xi.1 + .xi.2 e2

(2.15)

By similar calculations as for the lag 1 auto-covariance, we now find the lag 2 and lag 3 auto-covariances, namely, cov(lit, li,t-2) and cov(lit, li,t-3)·

Lag 2 auto-covariance:

cov(lit, li,t-2) =E(lit li,t-2) - E(lit) E(li,t-2)

=E[(ait li,t-1 + dit) li,t-2]- E(lit) E(li,t-2)

=E[{ait (ai,t-1 li,t-2 + di,t-1) + dit} li,t-2]- E(lit) E(li,t-2)

- E(lit) E(li,t-2)

>.7.1 1

/;,2

>.i.1 + >.i.2

= ( >.i.1 ) 2

>.i.1 + >.i.2 ).i.l + ^>.i.2 1;,2

2 >.i.1 + ^>.i.2

=pi 1;,2 ^(2.16)

Lag 3 auto-covariance:

cov(lit, li,t-3) =E(lit li,t-3)- E(lit) E(li,t-3)

=E[(ait li,t-1 + ^dit) li,t-3]- E(lit) E(li,t-3)

>.~.1 1

7,2 ( >.i.1 + ^>.i.2)2

= ( ).i.l ) 3

).i.l + ^>.i.2

>.i.1 + ^{>.i.2 /;,2}

(2.17) Note that by using (2.15), (2.16) and (2.17) and the formula for the correlation

( ) cov(lit, li t-t) .

given by corr Yit, Yi,t-l = ' , one obtams the lag 1, lag 2 and lag JVar(lit) Var(li,t-z)

3 correlation as Pi, pr and p~ respectively.

In the manner similar to these of lag 1, lag 2 and lag 3 correlation, we can obtain any lag correlations such as corr(Yiu, Yit) = p~-u, for u < t.

Aliter: We may have an alternative proof of the lemma by using general 't' as follows:

Note that

[

l-1

l

n

n

[

l-1

l [

l

cov(Yit, Yi,t-l) = cov (

n

n

[

l-1

l

= E IJ ai,t-j Var ( Yi,t-l I ait, ai,t-1, . .. , ai,t-l+1)

J=O

[

k-1 ^m-1

l

+ cov dt + ~ (

n

= E(ai,t-j)Var(Yi,t-l)

(2.19)

Hence the result.

Estimation of Parameters for Stationary Longitudinal Model

Recall from Section (2.2) that the AR(1) type gamma model is given by Yit = ait Yi,t-l + dit

where Yit marginally follows the gamma distribution denoted by Yit ^r-v Ga(>.i.l +.Ai.2 , ~),

with Ai.l = e-xUJ and >.i.2 = ( l-p;) Ai.l· Suppose that Pi = ). Ai.\ ^{= p for all}

p, i.l + ^i.2

i = ^{1, ... , K.} In this case, Ai.2 = ( 7") ^Ai.l· Note that it was shown in the last chapter that p~-u is the lag ( t-u) auto-correlation between Yiu and Yit ( u < t), Pi being the lag 1 correlation. When Pi = p is assumed, one deals with a stationary dynamic model with the same stationary auto-correlation structure for all individuals, which may be a reasonable situation in practice. In view of this, we consider p; = p for all i = 1, ... , K throughout the thesis.

It is clear from the above discussion that the statistical inference for the present gamma AR(1) model requires the estimation of the so-called regression effects /3,

the AR(1) auto-correlation parameter p and the scale parameter~· In the following subsection, we discuss the traditional maximum likelihood (ML) approach for the

12

estimation of these parameters but find that the ML approach is extremely complex from numerical point of view. As a remedy, we use the generalized quasilikelihood (GQL) approach suggested recently by Sutradhar (2003) which unlike the ML ap- proach requires only the first two moments of the data. The GQL approach provides consistent estimates for all parameters of the model and it appears to be much simpler as compared to the ML approach.

3.1 Likelihood Estimation and Its Complexity

3.1.1 Construction of the Likelihood Function

Note that the for a given i, the repeated responses for the ith individual, i.e., Yil, ... , Yit, ... , YiT are generated following the model (2.7). These responses are correlated, where the correlation structure is given by lemma (2.2). Let fi(Yil• ... , Yit• ... , Yiri,B, p, ~)

denote the joint density of the repeated gamma responses for the ith(i = 1, ... , K) individual. We then write the likelihood function under the model ( 2. 7) as

K

C(/3, p, ~) ⁼ II fi(Yil'" . 'Yit, " . 'YiT) (3.1)

i=1

=II K fi(Yi1) fi(YdYir) ... fi(YitiYi,t-d ... fi(YiT,Yi,T-1) (3.2)

i=1

where Yi1 ^I'.J Ga(.Ai.1 + Ai.2 , ~) leading the pdf

(3.3)

and fi(YitiYi,t-d denotes the conditional density of Yit given Yi,t-1, for t = 2, ... , T.

The derivation of this density function is given in the following lemma.

Lemma 3.1. For t = 2, ... , T, the conditional density of Yit given Yi,t-l under the model (2. 7) is given by

min(Yi,t-l,Yit)

f( I ) r(-Xi.I/P)

I

Yit Yi,t-1 = r(-\· ) [r(!::E,X. )]₂ c-(¥).\;.1 zt z,t-1

d p z.1 '> 0

(l=.e.).X· -1}

[(Yi,t-1-Zit)(Yit- Zit)] ^{P L1} dZit (3.4)

Proof. Using zit = ait Yi,t-b we first write the conditional distribution of zit given Yi,t-1 as

f (Zit I Yi,t-t) f(zit, Yi,t-d f(Yi,t-1) r(-Xi.l + -Xi.2)

r(-Xi.l) r(-Xi.2) ^{( )}

Ai.J-1 ( ) .\;.2-1

~ 1-~

Yi,t-1 Yi,t-1

1 (3.5)

Yi,t-1

Since Yit = ^Zit + dit, where dit = Yit - Zit ^I".J Ga(Ai.2, ~) and because Zit and dit are independent, by using (3.5), one may then write the conditional distribution of Yit given Yi,t-1 as

f(Yit I Yi,t-d = _{( )}

.\;.1-1 ( ) Ai.2-1

~ 1-~

Yi,t-1 Yi,t-1

1 Yi,t-1

(3.6)

Note that in (3.6), 0 < zit < min(Yit, Yi,t-1) as dit > zit ^=::} Yit > 0 and ~ < 1 ^=::}

Yi,t-1 Yi,t-1 > Zit·

After some algebra, this equation (3.6) yields

which by using Pi distribution

).i.l

- - - - = p for all i = 1, ... , K produces the conditional

>.i.l + >.i ^.2

min(yi,t-l,Yit)

f(Yit I Yi,t-1)

I

(!=.e.).A· -1}

[(Yi,t-1- zit)(Yit- zit)] ^{P '·1} dzit

for all t = 2, ... , T, as given in (3.4) mentioned in the lemma

3.1.2 Estimation

Let()= (/3', p, ~'· The ML estimation of() requires to solve the likelihood estimating equation ⁸¹~~ = 0 where the likelihood function £ is constructed in the previous

8log£

subsection. Note that the likelihood estimating equation

80 = 0 may be solved iteratively by using the Newton-Raphson equation

(3.7)

where iJML(r), for example, is the value of() obtained at the rth iteration. In order to compute (3.7), we need to calculate the first and second derivatives of the 'log£' with respect to '()'. By (3.2), the first derivative of the log-likelihood equation w.r.t

e may be written as

Note that () is a vector of three independent parameters, namely, /3, p and e. ^The

computations of :() log£, therefore, requires the calculations for :/3 log£,

:p

and 8

8e log£. The formulas for these derivatives are provided as follows. The actual

deduction of these results is however lengthy and complicated, which is deferred to the Appendix 1.

First order derivatives:

a a K KT 1 a

a{3log .C = a{3 I)og f(yil) _i=1 + L L _{i=1 t=2} ^{f( . I . _ ) a{3f(Yit I Yi,t-1)}

Yzt Yz,t 1

1 ^{K [} [r(Ai.t/ p)]'] ^{K T} 1

=--I:Ai.lxi. log~Yil- +2:2:---

p i=1 r(Ai.d p) i=1 t=2 f(Yit I Yi,t-1)

[

min(Yit,Yi,t-1) min(yit,Yi,t-1)

l

api

f f

X a{3 Qi Ti dzit +Pi I1i dzit ,

0 0

where

r(Ai.d p)

Pi= ¹

r(Ai.I) [r(7Ai.1)J2 C(7l"i.1

~

q . _ ^zAi.1-1 y P e-~(Yit-Zit) z - it i,t-1

Ti = [(Yi,t-1-Zit)(Yit- zit)(T·)Ai.l-1

aqi ari

fti = a{3 Ti + Qi a{3 · Next,

a a K KT 1 a

alog.C =a-I::logf(yil) + LL ^{!( . I . ) a f(Yit I Yi,t-1)}

p P i=1 i=1 t=2 Yzt Yz,t-1 P

K T

1

+ ~ ~ ^{f(Yit I Yi,t-1) [}

min(yit,Yi,t-1) min(yit,Yi,t-1)

l

api ^ap

f

f

0 0

(3.9)

(3.10)

(3.11) (3.12) (3.13)

(3.14)

where Pi, Qi and ri are as in (3.10), (3.11) and (3.12), and

(3.15)

Similarly,

8 8 ^{K K T} 1 8

8C log .c ^{= 8C} L log f(Yi1) + L L ^{!( . I . - ) 8Cf(Yit I Yi,t-d}

"' '> _{i= 1 i=1 t= 2} Y~t Yt,t 1 .,

K 1 ^{K K T} 1

= - L ^{Yi1 + -} L ^{Ai.1 +} L L . .

i=1 ~ P i=1 i=1 t=2 f(Yzt I Yz,t-1)

[

min(y;t,Yi,t-I) min(Yit.Yi,t-I)

l

x

:i ^j

j

0 0

(3.16)

where Pi, Qi and ri are as in (3.10), (3.11) and (3.12), and

(3.17)

Th ^/! 1 ^/! h d . . h 8pi 8qi 8ri 8pi 8qi 8ri 8pi 8qi d e 10rmu as 10r t e envatiVeS SUC as

813, 813,

813, 8p'

8p' 8p' 8e ' 8e an

~ required to compute (3.9), (3.14) and (3.16) are given in the appendix 1.

Second order derivatives:

To compute the second order derivatives, we note that

8² log £ 8² log £ 8² log £

a(3 Df3' 8!3 ap 8!3 a~

82

8() 8()' log .C = (3.18)

We now provide the formulas for the components of the second order derivative matrix in (3.18). The formula for the first leading component is given by

a^{2 log £} a^{2 K K T [ 1} a

a(3 a(3' = a(3a(3' I)og f(yil) + L L -^{]2( . I . _ ) a(3,f(Yit I Yi,t-1)}

i=₁ i=₁ t=2 Y~t Y~,t 1

KT[ ¹ a a 1

+ L L -^{]2( . I . ) 8(3,f(Yit I} Yi,t-1) 8(3f(Yit I Yi,t-d + J( . I . )

i=₁ t=₂ Y~t Y~,t-1 Y~t Y~,t-1

(3.19)

where, the formula for the Digamma function w( ·) and its derivative w'(·) may be found, for example, in Abramowitz and Stegun [1964, §6.3.1, p. 258; §6.4.12, p. 265]

which are given as r(z)' w(z) = r(z)'

1 1 1 1 1 1 1

W (z) rv-+- + - - - + - - - + ...

z 2z² 6z³ 30z⁵ 42z⁷ 30z⁹

The formulas for the remaining diagonal elements are given by

1 8² ]

+ ^{f( .} I . ) ~ ^2f(Yit I Yi,t-1) Yzt Yz,t-1 UP

1[

1 l

¹

- 2 - L ^{'11'(/\.d p) (} -2;\i.l)Ai.l + L L -^{j2( . I . )}

P i=1 P _{i= 1 t=2} Yzt Yz,t-1

min(yit,Yi,t-I) } {

2min(Yit,Yi,t-Il min(y;t,Yi,t-1)

( I )

I

I

+ Pi I2i dzit + f( . I . ) - a ₂ qi ri dzit + 2 -a I2i dzit

Yzt Yz,t-1 P P

0 0 0

(3.20)

and

1 a² J

+ J( . I . ) ac2f(Yit I Yi,t-d Yzt Yz,t-1 <,

respectively.

The formulas for the elements of the first off-diagonal of the matrix ( 3.18) are given by

a2 log .C a2 ^{K K T [} 1 a

a(3 a = fJ(3a L ^{log f(Yid +} L L -^{J2( .}

1

. _ ) a f(Yit ^I Yi,t-1)

P P i=1 i= 1 t=2 Y~t Y~,t 1 P

KT[ ¹ a a ¹

+ :L :L -^{J2( .} ₁ ^{. ) a f(Yit I} Yi,t-d a(3f(Yit ^I Yi,t-d + ^{!( .}

1

. )

i= 1 t=2 Y~t Y~,t-1 P Y~t Y~,t-1

and

a 1 a^{2 ]}

X 7 ) f(Yit I Yi,t-1) + !( . I . ) ^!;} !;}(:f(Yit I Yi,t-1) up Y~t Y~,t-1 up U<,

1 KT KT[ 1 a

=- C2 L L ^{Ai.l +} L L -^{j2( . I . ) acf(Yit I Yi,t-1)}

.,p i=1 t=2 i=1 t=2 Y~t Y~,t-1 "'

(3.23)

The elements of the second off-diagonal of the matrix (3.18) has the formula given by

a^{2log £} a^{2 K K T [ 1} a

a(3 a~ ^{= a(3ac} L ^{log f(Yi1) +} L L -^{J2( . I . - ) acf(Yit I Yi,t-1)}

"' i=1 i= 1 t=2 Ytt Yt,t 1 "'

1K KT[ ¹ a a

=- c L ^{Ai.1Xi. +} L L -^{J2( . I . - ) acf(Yit I} Yi,t-1) a(3f(Yit I Yi,t-1) ':.P _{i= 1 i= 1 t=2} Ytt Yt,t 1 "'

min(y;t,Yi,t-1) }]

J

+Pi a~ dzit .

0

(3.24)

It is clear from the above formulas (3.8)- (3.24) that the computation for the first

and second order derivatives necessary to construct the likelihood estimating equation

(3.8) is extremely cumbersome. This makes the maximum likelihood (ML) approach for the present longitudinal gamma AR(1) model ( 2. 7) practically less appealing. As a remedy, in the next section, we use a generalized quasilikelihood (GQL) approach suggested by Sutradhar (2003). Note that unlike the ML approach, this GQL ap- proach requires only the mean, variance and covariance of the longitudinal responses,

making the approach simpler and practically useful.

3.2 Quasi-Likelihood Estimation

Recall that the mean fLit = E(Yit), variance aitt = V(Yit), and the covariance aiut = E(Yiu - f.Liu) (Yit - J.Lit), for the repeated responses Yil, ... , Yit, . .. , YiT un- der the present gamma AR(1) model (2.7) were computed in section (2.2.1). Let Yi = (Yil, ... , Yit, ... , YiT )' be the T x 1 response vector for the ith individual, and f.Li = (Mil, ... , fLit, ... , fLiT)' and Ei = ( aiut) be the mean and covariance matrix of Yi, respectively. Here, J.Li and Ei are the functions of /3, p and e parameters. In the tradi- tional longitudinal set up, there exists a generalized quasilikelihood (GQL) approach [Sutradhar (2003)] to estimate the regression effects f3 consistently and efficiently, whereas other nuisance parameters are estimated consistently by using the method of moments. In this section, we follow this GQL approach and for known p and e, ^we

write the GQL estimating equation for f3 as

(3.25)

which may be solved iteratively by Newton-Raphson method using the iterative equa- tion given by

Snew =Sold+ ^(3.26)

Note that to construct the GQL estimating equation ( 3.25), it was assumed that p and e are known. But these parameters are rarely known in practice. As mentioned above, one may however obtain consistent estimates for these nuisance parameters by using the method of moments. The formulas for these estimates are given as in the following lemma.